Outer Limits of Reason (49 page)

 

Another way of seeing the mysterious relationship between science and mathematics is to look at a few instances where
entire fields
of mathematics developed long before physical applications of the fields were found.

Conic Sections and Kepler

The ancient Greeks loved geometry. In contrast to the ancient Egyptians, who used geometry to literally measure (
metron
) parts of the Earth (
ge
) for agricultural or legal reasons, the Greeks studied geometry purely for intellectual purposes. One of the brightest stars of Greek geometry was Apollonius (about 262–190 BC), who was born in Perga in southern Asia Minor. He studied what would happen if you take a cone and intersected it with a flat plane, as in
figure 8.2
.
²⁹

The curves that such intersections form are called
conic sections
. If the plane is lined up with the cone, the curve formed would be a circle. If the plane was slightly skewed, an ellipse would be formed. By further manipulating the plane, one can create parabolas and hyperbolas. Apollonius wrote a book about conic sections in which he stated about 400 theorems concerning the different properties of such curves.

Eighteen hundred years after Apollonius lived, Johannes Kepler (1571–1630) was trying to figure out how to make sense of Copernicus's radical new idea of having the sun in the center of the universe with the planets flying around the sun in giant circles. There was a terrible problem with Copernicus's new system: his predictions were wrong. The antiquated geocentric system of Ptolemy made better predictions than the new heliocentric system of Copernicus. Kepler realized that Copernicus's mistake was in thinking that the planets traveled around in circles. Rather, the orbits of the planets were ellipses. Because many of the properties of ellipses were worked out almost two millennia earlier, Kepler went back to study the ancient works of Apollonius to determine the properties of planetary motion. Once it was realized that the planets went in the well-understood elliptical motion, the positions of the planets were easy to predict. A historian of science wrote that “if the Greeks had not cultivated conic sections, Kepler could not have superseded Ptolemy.”
³⁰

How is it possible that the abstract writings of a long-dead Greek mathematician could help explain the motion of the planets?

Non-Euclidean Geometry and General Relativity

Classical Greek geometry found its most lasting form in the writings of Euclid (323–283 BC). His
Elements
was one of the most successful textbooks of all time. He began the book with ten obvious axioms. The first four are as follows:

1. A line segment can be drawn between any two points.

2. Any straight line segment can be extended to a straight line.

3. Given any straight line segment, one can draw a circle having the segment as radius and one endpoint as center.

4. All right angles are congruent (the same angle).

These laws are represented in
figure 8.3
.

Euclid's fifth axiom, which has come to be known as the
parallel postulate
, is worth careful study:

5. If two lines are drawn that intersect a third in such a way that the sum of the inner angles on one side is less than two right angles, then the two lines will inevitably intersect each other on that side if extended far enough.

This axiom is depicted in
figure 8.4
. In can be restated that if two lines are not parallel, they will eventually cross.

The fifth axiom has a different feel than the other axioms. The first four axioms are simple and easy to state. In contrast, although the fifth axiom is obviously true, it is more complicated and a little bothersome. The fifth axiom is the only axiom concerned with something (a crossing) that could occur some distance away from the angles under discussion. Euclid himself was suspicious of this axiom and tried to avoid using it as much as possible. Although mathematicians felt that the axiom was correct, they thought that it was more a consequence of the other axioms than an axiom itself. Throughout the centuries that Euclid's text was used, many people tried to derive the fifth axiom from the other nine axioms. That is, they tried to show the following derivation:

axioms 1–4 and axioms 6–10 ⇒ axiom 5.

They were unsuccessful. In 1767, French mathematician Jean-Baptiste d'Alembert (1717–1783) lamented the fact that for 2,000 years mathematicians could not prove the fifth axiom from the other axioms. He called this the scandal of geometry.

In the seventeenth century Girolamo Saccheri (1667–1733), Johann Heinrich Lambert (1728–1777), and Adrien-Marie Legendre (1752–1833) tried using a different method to prove that the fifth axiom was a consequence of the other axioms. Their method of proof is something that readers of this book are already comfortable with: a proof by contradiction. They tried to show that the fifth axiom is a consequence of the other axioms by assuming the other axioms and also assuming that the fifth axiom is
false
. Their goal was to show that this would to lead to a contradiction. In short:

axioms 1–4, 6–10 and the falsehood of axiom 5 ⇒ contradiction.

With this derivation in hand, we would have to say that the reason for the contradiction is that the truth of the fifth axiom is a consequence of the other axioms and that our assumption that the fifth axiom is false was wrong. But a funny thing happened on the road to proving this result: no contradiction could be found! These mathematicians went on and on deriving different strange theorems but could not find a clear contradiction. They were so confident that there must be some contradiction there that they “fudged their results” and found dubious artificial contradictions.

At some point, one of the greatest mathematicians of all time, Johann Carl Friedrich Gauss (1777–1855), and others realized that there is a reason why no one could find a clear contradiction: the fifth axiom can be true but it also can be false. That is, this complicated axiom did not depend on the others or was “independent” of the other nine axioms. When the parallel postulate is taken as true, the system studied is classical Euclidean geometry, and when it is taken as false, the system is called non-Euclidean geometry. Gauss never published anything on the topic so credit is usually given to the Hungarian Janos Bolai (1802–1860) and the Russian Nicolai Ivanovitch Lobachevsky (1793–1856) for founding this branch of mathematics. The topic was advanced by the German mathematician Bernhard Riemann (1826–1866).

Years later, Albert Einstein was looking for ways to express the ideas of general relativity. He was stuck. He could not find the proper language to describe the curves that space makes to influence the way matter moves (gravity.) Einstein's friend and teacher, Marcel Grossmann (1878–1936), suggested that he look into the abstract field of non-Euclidean geometry. To his shock, Einstein found exactly what he was looking for. The ideas and theorems of non-Euclidean geometry were precisely what he needed for general relativity. As Einstein wrote, “To this interpretation of geometry, I attach great importance, for should I have not been acquainted with it, I never would have been able to develop the theory of relativity.”
³¹
What are we to think of this? Here we have a bunch of mathematicians playing mind games with the usual axioms of geometry. They worked with an axiom system that used an “obviously” true axiom that was assumed to be false. Decades later that same system would miraculously help describe laws of the physical universe. Why should this be?

Abstract Algebra and Quantum Theory I: Complex Numbers

In high school, many students spend hours in math class learning to solve polynomial equations. Eventually they learn that some equations simply do not have a solution. The simplest such equation is

x
²
+ 1 = 0

For any
x
, we have that
x
²
is more than or equal to zero and adding 1 is definitely a positive number. So there is no
x
that could possibly satisfy this simple equation. In the sixteenth century, Gerolamo Cardano (1501–1576) posed the following question: What if we imagined that there was a solution to this problem? That is, let's work with a number
i
(for imaginary) and let this number be the solution to this simple equation. In other words, if

then plugging it into the equation, we get that

Obviously, this number cannot exist. But imagine that it does. We might then multiply this
i
by real numbers and create numbers like 2
i,
3
i, -
5.7
i
, etc. These are called
imaginary
numbers. Pressing on, we can add real numbers to imaginary numbers and get numbers like 7
+
3
i,
6.248 – 8.7
i
, or for any real numbers
a
and
b
we have
a
+
bi
. These numbers are a combination of both real and imaginary numbers and are called
complex
numbers. Mathematicians spent many long, lonely years working out many of the properties of these artificially manufactured numbers. All along physicists and others ignored these eccentric mathematicians and the strange curiosities they played with.